# Operator-state correspondence The operator-state correspondence, or state-operator correspondence, says that there is a one-to-one correspondence between states and local operators in a [[0481 Conformal field theory|CFT]]. This can be seen most easily in radial quantisation: a state at a given radius $r_f$ can be defined using the path integral by integrating over the annulus between some initial radius $r_i$ and $r_f$; taking the limit $r_i\to0$, the path integral then only depends on the local data at $r=0$, which can be captured by a local operator. ## In CFT - In CFT, relates a state in *highest weight rep* to a *local* operator in the manifold. - Use radial quantisation, inserting a primary at the origin creates a primary state. - Inserting not at the origin gives some linear combination of primary plus descendants. ## In BMSFT For BMS symmetry, states in *unitary rep* = insertions of local operators in super-momentum space. [[2019#Hijano]] - for highest weight reps (non-unitary) -> different operators ## Necessary conditions - There needs to be a bijective map between states of the Hilbert space and a Euclidean geometry preparing the state, but more importantly the Euclidean path integral needs to be compact. ([[2018#Belin, de Boer, Kruthoff]]) - In a QFT that is not a CFT, the argument for operator-state correspondence fails because, if we naively do the same and map things to a cylinder, there is no Hilbert space interpretation for the theory on the cylinder. This is mainly due to the fact that without parity symmetry (combined with inversion) one cannot define an inner product for the states on the cylinder